In this paper we present the operational properties of two integral transforms of Fourier type, provide the formulation of convolutions, and obtain eight new convolutions for those transforms. Moreover, we consider applications such as the construction of normed ring structures on L1(R), further applications to linear partial differential equations and an integral equation with a mixed Toeplitz-Hankel kernel. Mathematics Subject Classification (2000). Primary 42B10; Secondary 44A20, 44A35, 47G10.

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This paper gives a general formulation of convolutions for arbitrary linear operators from a linear space to a commutative algebra, constructs three convolutions for the Fourier transforms with geometric variables and four generalized convolutions for the Fourier-cosine, Fourier-sine transforms. With respect to applications, by using the constructed convolutions normed rings on L1(R n ) are constructed, and explicit solutions of integral equations of convolution type are obtained.

Abstract. In this paper, some results on the equi-boundedness of solutions, the stability of the zero and the existence of positive periodic solutions of nonlinear difference equation with variable delayare obtained. IntroductionThe properties of solutions of delay nonlinear difference equations have been studied extensively in recent years; (see for example the work in [1][2][3][4][5][6][7][8][9] and the references cited therein). In [1-2], we obtained some results for the asymptotic behaviour of solutions of nonlinear difference equations with time-invariant delay of the form Motivated by the work above, in this paper, we aim to study the equiboundedness of solutions, the stability of the zero and the existence of positive periodic solutions of nonlinear difference equation with variable delaywhere the functions α, λ are defined on the set of integers, the function m maps the set of integers to the set of positive integers, the function F is defined on the set of real numbers. In this paper, we denote by Z the set of integers, by Z + the set of positive integers and by R the set of real numbers.2000 Mathematics Subject Classification: 39A12.

Abstract. This paper proves some algebraic charaterizations and Volterra characterizations of the generalized right invertible operators of degree k in the case of k > 2. A class of equations induced by generalized right invertible operators of degree of k can be solved in a closed form.

Abstract. We deal with a class of integral equations on the unit circle in the complex plane with a regular part and with rotations of the formwhere T = M n1,k1 . . . M nm,km and M nj ,kj are of the form (3) below. We prove that under some assumptions on analytic continuation of the given functions, ( * ) is a singular integral equation for m odd and is a Fredholm equation for m even. Further, we prove that T is an algebraic operator with characteristic polynomial P T (t) = t 3 − t. By means of the Riemann boundary value problems, we give an algebraic method to obtain all solutions of equation ( * ) in closed form.

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In this paper we study a general equation in right invertible operator of order one in the ciwe when either resolving operator I-AR or 1-RA has a generalized almost inverse only. Moreover, we give the positive answer to the following question: Doed the left invertibility (right invertibility, invertibility) of 1-AR imply the left invertibility (right invertibility, invertibility) of the operator I-RA? (ef. [l], Open Question on p. 140). Let X be a linear space over a field 3 of scalars and let D E L ( X ) be right inver-tible, where by L ( X ) we denote the set of all linear operators with domains and ranges in X . Denote by L,(X) the set of all operators from L ( X ) which are defined on the whole space X. Let R ( X ) will stand for the set of all right invertible operators belonging to L ( X ) . For a D E R ( X ) we denote by ,RD the set of all its right inverses and by ZDthe set of all initial operators, i.c. the set We admit here and in the sequel that all right inverses belonging to L,(X) and that Consider the following equation ker D =t= ( O } , i.e. D is right invertible but not invertible.(1) (D -A ) s = y where A E L,(X), y 6 (D -A ) dom D. Definition 1 113. (i) The operators I-RA and I-AR are said to be resolwing operators /or the operator D -A . (ii) I / neither I-RA nor I-AR i s invertibze, then equation (1) & said to be ill-delermined. The solvability of ill-determined equations in the cases when resolving operators are either left or right invertible was studied by POQOIZZELEC [3]-[6] (cf. also [l]).